Exercise 6, § 7

Let K be a commutative field and E a vector space over K.

(a) For an element $z = \sum_{p=0}^\infty z_p$ (with $z_p \in \bigwedge^p E$) of the exterior algebra $\bigwedge E$ of E to be invertible, it is necessary and sufficient that $z_0 \neq 0$ (prove it first when E is finite-dimensional and then pass to the general case, noting that for all $z \in \bigwedge E$ there exists a finite-dimensional subspace F of E such that $z \in \bigwedge F$).

(b) Suppose that K is such that $2 \neq 0$ in K. Show that, if E is infinite-dimensional or of even finite dimension, the centre of the algebra $\bigwedge E$ consists of the elements such that $z_p = 0$ for all odd indices $p$. If E is of odd dimension $n$, the centre of $\bigwedge E$ is the sum of the above subspace and $\bigwedge^n E$. In all cases, the centre of $\wedge E$ is identical with the set of elements of $\wedge E$ permutable with all the invertible elements of $\wedge E$.